Timeline for Realizing continuum many types and omitting one
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 27, 2023 at 5:49 | comment | added | James E Hanson | Omitting types in uncountable languages is pretty subtle. I don't think it's going to be easy to find a relevant general theorem. | |
| Jul 27, 2023 at 1:58 | comment | added | Joel David Hamkins | But I don't agree that the theories are very similar. | |
| Jul 27, 2023 at 1:47 | comment | added | Joel David Hamkins | Thanks, that is helpful, and that is what I had thought you meant. | |
| Jul 27, 2023 at 1:39 | comment | added | TomKern | When I use "weak monadic second-order" here I'm just distinguishing between the theory of $(\mathbb{N}, \mathcal{F}(\mathbb{N}), \in, \leq)$ and $(\mathbb{N}, \mathcal{P}(\mathbb{N}), \in, \leq)$ where $\mathcal{F}$ is finite subsets and $\mathcal{P}$ is the full powerset. The theories are very similar, but only in the former do sets necessarily have largest elements, and the latter requires Buchi automata to analyze. On the other hand, my question is just about whether there is an omitting types-like theorem that I should be looking into. | |
| Jul 27, 2023 at 1:38 | comment | added | Joel David Hamkins | Yes, agreed. I just wanted to clarify the meaning of your $\zeta$, which is not as widely used that way these days as it was decades ago. It is a little old-fashined terminology, like using $\eta$ for the order type of $\mathbb{Q}$, whereas most people these days just write $\mathbb{Q}$. | |
| Jul 27, 2023 at 1:36 | comment | added | TomKern | Yes, but not just any model with order type $\omega+\zeta$ in the first order part (finding these is easy). I'm looking for a nonstandard model of the theory of $(\mathbb{N}, \mathcal{F}(\mathbb{N}), \in, \leq)$ with first order type $\mathbb{N} + \mathbb{Z}$ that also realizes the continuum many types specifying the $C_K$. | |
| Jul 27, 2023 at 1:34 | comment | added | Joel David Hamkins | Some people may not know that $\zeta$ you mean the order-type of the integers $\mathbb{Z}$. So you are looking for a model of order tyle $\mathbb{N}+\mathbb{Z}$. | |
| Jul 26, 2023 at 23:20 | comment | added | Joel David Hamkins | Thanks for clarifying. It seems you want a nonstandard model of the theory of $\langle\mathbb{N},F(\mathbb{N}),\in,\leq\rangle$ with order type $\omega+\zeta$ in the first-order part. Is that right? I find it odd to speak of a nonstandard model of a structure (which you do twice), but I guess you just mean a nonstandard model of the theory of that structure. It seems that "weak" means you only allow bounded monadic classes into the second-order part—is that right? | |
| Jul 26, 2023 at 23:04 | history | edited | TomKern | CC BY-SA 4.0 |
added 1544 characters in body
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| Jul 26, 2023 at 22:13 | comment | added | Joel David Hamkins | Can you say a little more about what exactly the theory is? Do you want your model to be an elementary extension of the full monadic second-order standard model? | |
| Jul 26, 2023 at 21:31 | history | asked | TomKern | CC BY-SA 4.0 |